Finite Group Databasebeta

← #28312254

Subgroups — #28312254

Order 13568.

342 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290916C211yesmaximal, characteristic
2#20290656C211yesmaximal, characteristic
3#20289776C214yesmaximal
4#20285228C211yesmaximal, characteristic
5C2×D53⋊C16C2212yes—
6C2×#13133288—21no—
7C2×#13133288C2212yes—
8#16811627—21no—
9#16811512—21no—
10#16811511—21no—
11#16811179—21no—
12#16811027C412yes—
13#16811025C411yescharacteristic
14#16811022—21no—
15#16811020—21no—
16#16810660C2212yes—
17#16810658—21no—
18C4×C8×D53C2211yescharacteristic
19C53×#499C411yescharacteristic
20C2×#11719936—22no—
21D53⋊C16—22no—
22D53⋊C16Q814yescomplemented
23C2×#11719910—22no—
24#13133288—24no—
25#13133288—42no—
26C4⋊#88324—42no—
27C2×C8×D53D411yescharacteristic
28C2×C8×D53C2311yescharacteristic
29#13133216—22no—
30#13133209—21no—
31#13133195—42no—
32#13133182—42no—
33#13133175—22no—
34C42×D53D411yescharacteristic
35#13133151C2×C412yes—
36#13133150—22no—
37#13133149—21no—
38#13133148—22no—
39C4×#88322D411yescharacteristic
40C8×Dic53C2×C411yescharacteristic
41C8×Dic53D411yescharacteristic
42#125×C53—21no—
43#107×C53C2×C412yes—
44#106×C53—21no—
45C4×C8×C53C2×C411yescharacteristic
46C2×C4×D53SD1612yes—
47C2×C4×D53C2×D411yescharacteristic
48C2×Dic106—42no—
49C2×#88324—42no—
50#11719936—42no—
51#11719936—42no—
52C2×C8×C53C22⋊C411yescharacteristic
53C2×C8×C53C22×C411yescharacteristic
54C4⋊C4×C53—22no—
55C4⋊C4×C53—42no—
56C42×C53C22⋊C411yescharacteristic
57#11719921—42no—
58C4⋊Dic53—22no—
59#11719919—42no—
60#11719919—42no—
61C4×Dic53—42no—
62C4×Dic53C22⋊C411yescharacteristic
63C4×Dic53D812yes—
64C2×#88322C22⋊C411yescharacteristic
65C2×#88322C2×D411yescharacteristic
66C8×D53SD1612yes—
67C8×D53Q1612yes—
68C8×D53C2×Q812yes—
69C8×D53#4112yes—
70#11719910—24no—
71C22×D53C2×SD1611yescharacteristic
72C2×C4×C53—42no—
73C2×C4×C53#10212yes—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×Dic53—42no—
76C2×Dic53#10212yes—
77C2×Dic53C2×D811yescharacteristic
78C4×D53—42no—
79C4×D53#12411yescharacteristic
80C4×D53C4⋊Q811yescharacteristic
81C4×D53C2×SD1611yescharacteristic
82C4×D53C2×Q1611yescharacteristic
83Dic106—84no—
84#88324—82no—
85C8×C53#10312yes—
86C8×C53C4×Q812yes—
87#88322#10312yes—
88#88322C22⋊Q812yes—
89#13286—531nomaximal, Sylow
90C22×C53C2×#10211yescharacteristic
91C2×D53—22no—
92C2×D53#48711yescharacteristic
93C2×D53#49811yescharacteristic
94C4×C53—22no—
95C4×C53—82no—
96C4×C53#39111yescharacteristic
97C4×C53C2×#10311yescharacteristic
98Dic53—22no—
99Dic53—82no—
100Dic53#48611yescharacteristic
101Dic53#50011yescharacteristic
102#3160—531no—
103C2×#346—531no—
104#1848—531no—
105#1382—534no—
106C2×C53#199011yescharacteristic
107C2×C53#214312yes—
108D53—21no—
109D53—21no—
110#533—1061no—
111C22×C16—532no—
112#500—532no—
113#499—531no—
114#499—1061no—
115#499#615711yescharacteristic
116#498—1061no—
117C2×#108—532no—
118C2×#108—1061no—
119#427—532no—
120#427—1061no—
121#423—1061no—
122C2×C4×C8—531no—
123#346—1061no—
124#346—1061no—
125C53#1328611yescharacteristic, complemented, Sylow
126C2×M16—1062no—
127C22×C8—531no—
128C22×C8—531no—
129C4⋊Q8—1062no—
130#125—21no—
131#125—1061no—
132C22⋊Q8—2122no—
133C4×Q8—2122no—
134#117—1062no—
135C2×C42—531no—
136C2×C16—534no—
137C2×C16—1062no—
138C2×C16—1062no—
139#108—1064no—
140#108—2122no—
141#107—532no—
142#107—1062no—
143#107C2×#615712yes—
144#106—21no—
145#106—1062no—
146#106—1061no—
147C4⋊C8—2122no—
148C4×C8—531no—
149C4×C8—531no—
150C4×C8—531no—
151C4×C8C2×#615711yescharacteristic
152C2×Q8—2122no—
153C22×C4—531no—
154C22×C4—532no—
155M16—2122no—
156M16—2122no—
157C2×C8—531no—
158C2×C8—531no—
159C2×C8—532no—
160C2×C8—532no—
161C2×C8—532no—
162C2×C8—532no—
163C2×C8—2122no—
164C2×C8C22⋊#615711yescharacteristic
165C2×C8C22×#615711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—1062no—
169C4⋊C4—2122no—
170C4⋊C4—2122no—
171C22⋊C4—2122no—
172C42—532no—
173C42—531no—
174C42—2122no—
175C42C22⋊#615711yescharacteristic
176C16—1064no—
177C23—531no—
178Q8—4244no—
179C2×C4—42no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—531no—
185C2×C4—532no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4#1313327912yes—
189C2×C4C2×C22⋊#615711yescharacteristic
190C8—532no—
191C8—532no—
192C8—4242no—
193C8#1313328012yes—
194C8Q8×#615712yes—
195C22—531no—
196C22—531no—
197C22—1062no—
198C22C2×#1313327911yescharacteristic
199C4—22no—
200C4—82no—
201C4—531no—
202C4—531no—
203C4—1062no—
204C4—4242no—
205C4(C2×#6157)⋊Q811yescharacteristic
206C4C2×#1313328011yescharacteristic
207C2—1061no—
208C2—1061no—
209C2#2029089611yescharacteristic
210C2#2029102912yes—
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#499C4
socleC22×C53C2×#102
prime radicalC22×D53C2×SD16
cosocleC2×C8×D53C23
hypercenter#499#6157
hypocenterC53#13286
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #499

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
22. D53⋊C16Q8semidirect
125. C53#13286semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202909162
#202906562
#202897762
#202852282
#1328653